Abstract

A triparametric family of fourth-order multiple-zero solvers have been proposed. In this paper, we select among them a uniparametric family of optimal fourth-order multiple-zero solvers with rational weight functions and pursue their dynamics by exploring the relevant parameter spaces and dynamical planes, by means of Möbius conjugacy map applied to a prototype polynomial of the form . The resulting dynamics is best illustrated through various stability surfaces and parameter spaces as well as dynamical planes.

1. Introduction

The root-finding problem [1] plays a significant role in many branches of computational sciences. It arises in various fields such as physics, biochemistry, applied engineering, and earth sciences, including industrial mathematics. A considerable number of literatures [28] can be found describing the dynamical behavior of iterative methods to locate the multiple root of a nonlinear equation under consideration.

Many of such existing literatures have studied and emphasized local convergence behavior of the iterative root-finders for nonlinear governing equations under consideration through the viewpoints of the relevant basins of attraction. It is, however, not only worthwhile but also important to investigate the convergence behavior of the root-finders in a global sense. Dealing with such a global convergence behavior certainly motivates our current investigation via the concept of the parameter space where the relevant dynamics of the iterative root-finders under their free critical points continuously evolves as the values of parameter vary along the axes of the extended complex plane.

Homeomorphic conjugacy maps will be introduced in Section 2 in order to better understand the dynamics of the given iterative zero solvers in view of the fact that a topologically conjugated dynamical system preserves its orbit behavior as well as fixed point properties. As a convenient homeomorphic conjugacy map, we will make use of Möbius conjugacy map that enables us to effectively treat the relevant dynamics, by observing that the inverse of Möbius conjugacy map is also of Möbius-type. Indeed, Theorem 3 will show the desired resulting conjugacy map to be studied under current investigation. Additional results on the dynamical behavior including parameter spaces and dynamical planes will be shown in later sections.

To proceed with our investigation, we will employ the triparametric family of fourth-order multiple-zero methods developed by Kim and Geum [9] and introduce a uniparametric family of fourth-order multiple-zero solvers with rational weight functions as follows: where is a multiplicity of the sought zero, ,  ,  ,  , and and is a free parameter.

The aim of this paper is to investigate the complex dynamics on the Riemann sphere by analyzing the parameter spaces associated with the free critical points and the dynamical planes related to the uniparametric family of fourth-order multiple-zero solvers. Such investigation from a viewpoint of complex dynamics may have a drawback restricting us from treating the real dynamics for real nonlinear equations. Nonetheless, our primary motivation for analyzing the relevant complex dynamics lies in seeking the dynamical behavior of a family of iterative methods (1) via Möbius conjugacy map by presenting -parameter spaces and the corresponding dynamical planes.

The rest of this paper is made up of three sections. In Section 2, conjugacy maps along with the property of dynamical analysis for the aforementioned numerical methods are studied and the stability surfaces of the strange fixed points for the conjugacy map are displayed. Section 3 shows the relevant complex dynamics including the parameter spaces and the dynamical planes associated with the basins of attraction. In the last section, we draw a conclusion and suggest the future study by extending the current analysis.

2. Conjugacy Maps and Dynamics

A nonlinear equation (1) can be written in a generic form as a discrete dynamical system:where is the iteration function. As a result, we obtain a complex discrete dynamical system: where ,  , and .

The following definition and remark are useful to construct the conjugacy map and to investigate the relevant dynamics.

Definition 1. Let and be two functions (dynamical systems). We say that and are conjugate if there is a function such that . Then the map is called a conjugacy [10].

Remark 2. Note that a conjugacy indeed preserves the dynamical behavior between the two dynamical systems; for example, if is conjugate to via and is a fixed point of , then is a fixed point of .
Furthermore, if is a homeomorphism, that is, if is topologically conjugate to via , and is a fixed point of , then is a fixed point of . Also, we find and = . If and are invertible, then the topological conjugacy maps an orbit of onto an orbit of , where and the order of points is preserved. Hence, the orbits of the two maps behave similarly under homeomorphism or .

Via Möbius conjugacy map with ,  ,  considered by Blanchard [11], in (3) is conjugated to satisfyingwhen applied to a quadratic polynomial raised to the power of , where and are polynomials with no common factors whose coefficients are generally dependent upon parameters , , and . The following theorem shows that is dependent only on but independent of parameters and .

Theorem 3. Let with and . Then is conjugate to satisfying where

Proof. Since the inverse of is easily found to be , we find after a lengthy computation with the aid of Mathematica [12] symbolic capability: where and are polynomials of degree at most in with a single free parameter . This gives the desired result, completing the proof.

The result of Theorem 3 enables us to discover that (corresponding to fixed point of or root of ) and (corresponding to fixed point of or root of are clearly two of their fixed points of the conjugate map , regardless of -values. Besides, by direct computation, we find that is a strange fixed point [1315] of (that is not a root of ) due to the fact that , regardless of -values.

We now seek further strange fixed points including (corresponding to the original convergence to infinity in view of the fact that or ). To do so, we first investigate some properties of stated in the following theorem.

Theorem 4. Let and be given by (9). Then the following hold.
(a) The leading highest-order term of is given by , provided that .
(b) has a factor , provided that .
(c) , and , where with and .
(d) approaches as tends to , provided that .

Proof. After a lengthy computation and careful algebraic treatments with the aid of Mathematica, (a), (c) follow without difficulty. For the proof of (b), we directly compute the values of and . The proof of (d) follows from the fact that , by using (a) along with a highest-order term of having degree at most .

We will find out the fixed points of the iteration function . Let , whose zeros are the desired fixed points of . From (b) and (c) of Theorem 4, we find that and are the roots of . Hence the expression of will take the following form: whereand are given in Theorem 3.

As a result, ,  , and are the fixed points of . Among these fixed points, is a strange fixed point that is not root or . Further strange fixed points are calculated from the roots of . The following theorem describes some properties of .

Theorem 5. Let be given by (10). Then the following hold.
(a) for , regardless of -values.
(b) has double roots at and , that is, has a factor , provided that for and for .
(c) for , regardless of -values, where ,  .
(d) has also double roots at and , that is, has a common factor as shown in , provided that for and for .
(e) , for , regardless of -values.

Proof. Via careful algebraic treatments and symbolic computation with the aid of Mathematica, (a), (c), (e) follow without difficulty. For the proof of (b), we directly compute the values of and In view of the relations, ,  ,  . We find and for with and with . The proof of (d) follows from the fact that and We also find and

With the use of properties of , we now consider some strange fixed points along with their stability for selected values of and .

To continue our investigation of dynamics behind iterative map (3) applied to a quadratic polynomial raised to the power of ,  , we will describe the fixed points of in (9) and their stability. In view of the fact that is a fixed point of for a fixed point of with , we are interested in the explicit form of for as follows:where we conveniently denote

This enables us to discover that (corresponding to fixed point of or root of ) and (corresponding to fixed point of or root of are clearly two of their fixed points regardless of . To find further strange fixed points, we solve in (15) for with typical values of .

We now investigate further strange fixed points including (corresponding to the original convergence to infinity in view of the fact that or ). By direct computation, we will describe the roots of for . To this end, we first check the existence of -values for common factors (divisors) of and . Besides, will be checked if it has a divisor or . The following theorem best describes relevant properties of such existence as well as explicit strange fixed points.

Theorem 6. Let in (15). Then the following hold.
(a) If , then and the strange fixed points are given by and .
(b) If , then and the strange fixed points are given by and .
(c) If , then and the strange fixed points are given by and ,  .
(d) Let . Then holds for . Hence, if is a root of , then so is .

Proof. (a)–(c) Suppose that and for some values of . Observe that parameter exists in a linear fashion in all coefficients of both polynomials. By eliminating from the two polynomials, we obtain the relation: . Hence, any root of is a candidate for a common divisor of and . Substituting all the roots of into and , we find required relations for and, solving them for , we find . The remaining part of the proof is straightforward. (d) If is a divisor of , then yielding , which is already handled in (b). If is a divisor of , then , yielding . Then remaining proof is trivial. (e) By direct substitution, we find without difficulty. Hence if and only if for . This completes the proof.

Theorem 7. Let in (15). Then the following hold.
(a) If , then and the strange fixed points are given by and .
(b) If , then and the strange fixed points are given by ,  , and .
(c) If , then and the strange fixed points are given by ,  ,  ,  , and .
(d) Let . Then holds for . Hence, if is a root of , then so is .

Proof. The proofs immediately follow from the same argument as used in the proofs of Theorem 6.

As a result of Theorem 5(a), we find the fixed points of , that is, the roots of explicitly as stated in the following corollary.

Corollary 8. Let be a root of , that is, a root of for in (15). Suppose and have no common factors for some suitable -values. Then the roots of for are explicitly given by the following.
(a) The four roots of are explicitly found to be (b) The eight roots of are explicitly found to be where

Proof. Since is a root of for , so is from the result of Theorem 5(a). For the proof of (a), thus can be written as a product of two factors: By expanding the right side of the above equation and comparing the coefficients of the first and second-order terms, we find two relations: which gives the desired values of , . Then the four roots can be found explicitly from or . Similarly for the proof of (b), can be written as a product of four factors: By the same argument as used in the proof of (a), the desired result follows. This completes the proof.

We find that can be reduced to a fraction of a common denominator as follows: where is a polynomial of degree at most defined bywith ,   and ;   and are described earlier in Theorem 6.

We are now ready to determine the stability of the fixed points. In particular, it is necessary to compute the derivative of from Theorem 19: where

We first check the existence of -values for common factors (divisors) of and . Besides, will be checked if it has divisors and . The following theorem best describes relevant properties of such existence as well as explicit strange fixed points.

Theorem 9. Let in (25). Then the following hold.
(a) If , then .
(b) If , then .
(c) If , then .
(d) If , then .
(e) If , then .
(f) If , then .
(g) Let . Let be a fixed point of satisfying . Then holds for .

Proof. The proofs of (a)–(f) immediately follow from the same argument as used in the proofs of Theorem 19. Eliminating from the two polynomials and plays a key role in obtaining the relation: , whose roots enable us to deduce some desired -values. Additional requirement that are candidates for common divisors of and gives only . For the proof of (g), we use Theorem 19 to find , where . This completes the proof.

Theorem 10. Let in (25). Then the following hold.
(a) If , then .
(b) If , then .
(c) If , then , where
.
(d) Let . Let be a fixed point of satisfying . Then holds for .

Proof. The proofs of (a)–(c) immediately follow from the same argument as used in the proofs of Theorem 19. (d) We use Theorem 19 to find , where + . This completes the proof.

Table 1 summarizes the stability results for the strange fixed points of for special -values with .

We are ready to discuss the stability of the fixed points described in Theorems 6 and 7 in terms of parameter .

Theorem 11. Let and . Then the following hold.
(a) The strange fixed point becomes an attractor, parabolic (indifferent, neutral) point, and a repulser, respectively, when ,  , and .
(b) The strange fixed point is a superattractor if .

Proof. (a) From the case of in (25), we find . Solving for , we obtain circle in the cross-sectional -parameter plane for to be a parabolic point, where and . (b) Solving easily yields .

Theorem 12. Let and . Then the following hold.
(a) The strange fixed point is a parabolic (neutral, indifferent) point, respectively, when ,  , and .
(b) The strange fixed point is a superattractor if .

Proof. From the case of in (25), we find . Solving for , we obtain an ellipse in the cross-sectional -parameter plane for to be a parabolic point, where and . (b) Solving easily yields .

We now proceed to discuss the stability of the strange fixed points for conjugate map with using . As a consequence of Theorems 9(g) and 10(d) together with Corollary 8, the stability can be stated at most five strange fixed points including . Then the stability of these fixed points can be best described by illustrative conical surfaces shown in Figures 1-2. The top row of each figure refers to a stability surface for strange fixed point . The stability surfaces for the remaining fixed points are displayed in order from top to bottom and from left to right in each case of and . The underlying theory is clearly verified via cross-sectional views of the stability surfaces with -parameter domains.

The critical points of the iterative method are given by the roots of . Clearly, and are critical points associated with the roots and of the polynomial . The critical points that are not related to any roots of the polynomial as free critical points.

Remark 13. In view of (25), the following hold.
(i) When and , four roots of are the free critical points given by where
(ii) When and ,  12 roots of can be found numerically for a given .

For Remark 13(i), we further find that since is a root of , so is from Theorem 19. Hence, there exist two constants and such that For convenience, we define a function Then for satisfying , 4 roots of are the free critical points of ,  ,  ,  ,  , where

For Remark 13(ii), we also find that if is a root of , then so is from Theorem 19. Hence there exist six constants such that With the use of function introduced earlier, all 12 roots of can be written as Then desired among six constants can be found by comparing coefficients of six terms up to order 6 with those of in (25); eliminating from these relations, we find which constitutes a sextic equation in . For simplicity, we denote the left side of (31) by . Then the six roots of are found to the desired six constants. By solving numerically for a given , we can determine the desired six constants . Having found such , the six critical points can be given by for .

3. Parameter Spaces and Basins of Attraction

It is interesting to study the relevant complex dynamics from the viewpoint of parameter spaces and dynamical planes.

3.1. Parameter Spaces

It is our further interest to investigate the relevant complex dynamics from the viewpoint of parameter spaces and dynamical planes. The following lemma will be used for us to claim the favorable properties of symmetry on both parameter spaces and dynamical planes.

Lemma 14. Let be defined by , where are complex polynomials with real coefficients. Suppose . Let denote a complex conjugate of . Then the following hold.
(a) .
(b) If is a root of , then so is .

Proof. (a) By directly solving for by means of , we findSince are complex polynomials with real coefficients, we get
In view of (32), On the other hand, we also have implying .
(b) Let be a root of . Then stating that is also a root of .

Theorem 15. Let be a free critical point of dependent upon parameter given by a root of described in (15). Then the corresponding parameter space is symmetric with respect to its horizontal axis.

Proof. It suffices to show the claim for . We find that such is a root of as shown in (15) for a given . Then is also a root of (i.e., a free critical point of ) at from Lemma 14. For such a free critical point , consider conjugated map from (9): This expression allows us to get which states that the magnitude of the orbit of free critical point at is that same as that of the orbit of free critical point at and hence implies that the parameter space associated with map is symmetric with respect to its horizontal axis. The proofs for are similar.

Theorem 16. Given a parameter , let be a starting point of iterative map described in (15). Then the corresponding dynamical plane is symmetric with respect to its horizontal axis.

Proof. We first note that and similarly follow the proof Theorem 15. We now consider : which states that the magnitude of the orbit of is the same as that of the orbit of and hence implies that the dynamical plane associated with map is symmetric with respect to its horizontal axis if is real.

We shall describe further properties on parameter spaces and dynamical planes associated with conjugated map . By direct computation from in (9), we obtain the following lemma.

Lemma 17. for any and .

Corollary 18. Let be given. If is a -periodic point of , then so is .

Proof. As a result of Lemma 17 and via induction on , we find that holds for any integer and for any and . Hence , stating that is also a -periodic point of .

Theorem 19. Let be a critical point. Then the following holds.
(i)
(ii) If is a critical point, then so is .

Proof. (i) Since and is a critical point satisfying , we find that using (23).
(ii) Hence implies This completes the proof.

From (25), we first conveniently let and denote the branches of the free critical points of by in order. Indeed, we find that, for if and for if , where ; are described after Remark 13. For instance, when , we consider the orbit behavior of two branches and of the free critical points under the action of . The orbit behavior of other two branches and can be similarly described and hence its investigation will be omitted here.

The following remark is useful in understanding the orbit behavior of two branches that are reciprocals of each other among the four free critical points of .

Remark 20. In view of Lemma 17 and Corollary 18, we find that for holds for any integer and . Hence, the critical orbit of one branch behaves in quite the same way as the other branch does in the following sense.
(1) If the critical orbit of one branch converges to a -cycle with an integer , then so does the other branch.
(2) If the critical orbit of one branch is divergent but bounded, then so is the other branch.
(3) If the critical orbit of one branch converges to , then the other branch converges to , and vice versa.
Therefore, we find that the -parameter space associated with one branch has same components [16] and boundaries in a neighborhood of which different anomalies occur as the other branch does. It is interesting to observe that the component associated with fixed point shares its boundary with the component associated with fixed point , in each branch.

The case when can be similarly treated.

Without loss of generality, we limit ourselves to considering the case when , since similar treatment applies to the case when . In view of the above remark, it suffices to consider two branches and for their orbit behavior. Now, we are going to look for the best members of the family by means of the parameter space associated with the free critical points or .

Let = a critical orbit of under tends to a number . Similarly, we let = . We call and , respectively, as the parameter space and the dynamical plane (showing long-term orbit behavior) associated with . If the number or is a finite constant, then there exist finite periods in the orbit. Otherwise, the orbit is nonperiodic but bounded or goes to infinity. One should find that can be treated as a fixed point in the dynamics on the Riemann sphere.

We now conveniently introduce a systematic scheme coloring a point or point based on the period of the orbit of under the action of as follows. Let be a point in or . Then is painted in specified color if induces a -periodic orbit with under the action of . We accept the desired -periodic convergence of an orbit associated with or after a maximum of 2000 iterations and with a tolerance of . We further identify color according to the scheme shown in Table 2.

In Figures 3-4, parameter spaces associated with are shown. A point in is painted according to the coloring scheme defined in Table 2. One should note that every point of the parameter space whose color is not cyan (root ), magenta (root ), yellow, and red is not a good choice of in terms of relevant numerical behavior.

For convenience of analysis, we now let denote the parameter space associated with branch for .

Focusing the attention on the regions shown in Figures 314, it is evident that there exist members of the family with complicated behavior. We clearly see from Figure 11 that there exist components where -periodic orbits are generated with budding from period- component (in red color) and 4-periodic orbits budding from period- component (in orange color). We further observe that -periodic components with can be evidently seen as indicated by arrow lines. It is interesting for us to watch these components budded along the boundary of the period- component in the manner of Farey sequence [17], according to the Schleicher’s Algorithm [18].

We have painted the points both in the parameter space and in the dynamical plane with the same coloring scheme as defined in Table 2. In Figure 13, the dynamical planes of members of the family with regions of convergence to any of the strange fixed points are shown for the values of , respectively. Some interesting cycles of finite period are clearly seen. In Figures 13, we observe the dynamical planes of a member of the family with regions of convergence to -periodic cycles for in order from left to right and from top to bottom.

In summary, when , we have thus far observed that there exist regions of finite period for stable -cycles with . We also observe the fascinating fractal boundaries between the basins of attraction associated with different cycles.

For the remaining case of , we have carried out similar analysis to explore the relevant dynamics described in subsequent paragraphs.

Figures 510 show the parameter spaces associated with for . We evidently observe period-doubling components from Figures 59. For instance, Figures 5(c) and 6(c) exhibit components with periodic-doubling sequences and , respectively.

It is interesting to note in Figure 7(c) that -periodic components () arise along the boundary of a 1-periodic component in yellow where any -value can induce an orbit in the dynamical plane converging to a fixed point . The -periodic components here are not likely to occur in the manner of Farey sequence. It is worthwhile to note that both Figures 7(d) and 8(c) contain Mandelbrot-like components. By magnifying Figure 8(b), we obtain Figure 12 leading to Farey-sequential components along the boundary of a red component for .

Figure 14 displays dynamical planes associated with when , with various values of inducing -periodic cycles for in order from left to right and from top to bottom.

4. Conclusion

Given multiplicity , we have developed a uniparametric family of optimal fourth-order multiple-zero solvers with rational weight functions and investigated their complex dynamics via Möbius conjugacy map applied to a polynomial of the form along with the stability analysis of strange fixed points. The relevant dynamical analysis has been carried out from the viewpoint of stability analysis and in terms of parameter spaces and dynamical planes associated with basins of attraction. In the current approach, although we have encountered technical difficulties in explicitly finding critical points from (25) when in terms of parameter , a numerical approach developed based on Remark 13(ii) has been employed to resolve such technical difficulties. With the use of continuous dependence of zeros of a polynomial on its coefficients, that is, on -parameter values, it has been possible to construct numerical parameter spaces associated with twelve roots of in (25). Nevertheless, such continuous dependence on -parameter is not globally guaranteed since a numerical solution along one solution branch may trace along another nearby solution branch. This unfavorable result would be inevitable unless each solution branch is exactly found.

A future study dealing with the dynamics of other types of iterative methods will be primarily concerned with exact free critical points in order to construct favorable parameter spaces via global continuity of -parameter.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

The corresponding author (Young Hee Geum) was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education under the research grant (Project no. 2015-R1D1A3A-01020808).